MATHEMATICAL COMMUNICATIONS
Math. Commun., Vol. 15, No. 1, pp. 189-204 (2010)
189
Skew CR-warped products of Kaehler manifolds
Bayram S.ahin1,∗
1
Department of Mathematics, Inonu University, Malatya-44 280, Turkey
Received June 4, 2009; accepted January 2, 2010
Abstract. Warped product CR-submanifolds of Kaehler manifolds were introduced by
Chen in [9]. In this paper, we introduce a warped product skew-CR-submanifold, which is
a generalization of warped product CR-submanifolds. We give a characterization supported
by an example and obtain an inequality in terms of the length of the second fundamental
form of such submanifolds. The equality case is also investigated.
AMS subject classifications: 53C40,53C42,53C15
Key words: warped product manifold, CR-submanifold, slant submanifold, skew CRsubmanifold.
1. Introduction
CR-submanifolds were defined by Bejancu [1] as a generalization complex and totally
real submanifolds. A CR-submanifold is called proper if it is neither complex nor
totally real submanifold. The geometry of CR-submanifolds has been studied in
several papers since then.
Another generalization of complex (holomorphic) and totally real submanifolds
is a slant submanifold. Slant submanifolds were defined by Chen [8]. Since then,
such submanifolds have been investigated by many authors (see, [8] and references
therein). A slant submanifold is called proper if it is neither holomorphic nor totally
real submanifold and notice that a proper CR-submanifold is never a slant submanifold. In [21], Papaghiuc gave a generalization of this notion defining semi-slant
submanifolds, obtaining slant and CR-submanifolds as particular cases. We note
that slant and semi-slant submanifolds of Sasakian manifolds were studied in [4] and
[5] by Cabrerizo, Carriazo, Fernandez and Fernandez. On the other hand, in [25]
we studied hemi-slant submanifolds which were defined by Carriazo under the name
of anti-slant submanifolds [6] and showed that such submanifolds also contain CRsubmanifolds and slant submanifolds as particular subspaces. Moreover, we observe
that there is no inclusion relation between semi-slant submanifolds and hemi-slant
submanifolds.
In [22], Ronsse introduced skew CR-submanifolds of Kaehler manifolds. It is easy
to observe that a skew CR-submanifold is a generalization of slant submanifolds
as well as CR-submanifolds. In fact, semi-slant submanifolds [21] and hemi-slant
submanifolds [25] are also particular classes of skew CR-submanifolds.
∗ Corresponding
author. Email address: [email protected] (B. S.ahin)
http://www.mathos.hr/mc
c
°2010
Department of Mathematics, University of Osijek
190
B. S.ahin
Let (B, g1 ) and (F, g2 ) be two Riemannian manifolds, f : B → (0, ∞) and
π : B × F → B, η : B × F → F the projection maps given by π(p, q) = p and
η(p, q) = q for every (p, q) ∈ B × F. The warped product M = B × F is the manifold
B × F equipped with the Riemannian structure such that
g(X, Y ) = g1 (π∗ X, π∗ Y ) + (f oπ)2 g2 (η∗ X, η∗ Y )
for every X and Y of M and ∗ is a symbol for the tangent map. The function f
is called the warping function of the warped product manifold. In particular, if the
warping function is constant, then the manifold M is said to be trivial. Let X, Y be
vector fields on B and V, W vector fields on F , then from Lemma 7.3 of [2], we have
∇X V = ∇V X = (
Xf
)V
f
(1)
where ∇ is the Levi-Civita connection on M . We note that the concept of warped
products was first introduced by Bishop and O’Neill [2] to construct examples of
Riemannian manifolds with negative curvature. We also note that warped product
manifolds have their applications in general relativity. Indeed, many spacetime models are examples of warped product manifolds. More precisely, Robertson-Walker
spacetimes, asymptotically flat spacetimes, Schwarzschild spacetimes and ReissnerNordström spacetimes are examples of warped product manifolds, for details [17].
It is known that there are two distributions on a CR-submanifold such that one of
them is invariant and the other one is anti-invariant under the action of the complex
structure of the ambient space. It is also known that a warped product manifold
has fibres and leaves. Using this similarity, Chen considered warped product CRsubmanifolds of Kaehler manifolds [9], [10] and showed that there do not exist warped
product CR-submanifolds in the from M⊥ ×f MT such that MT is a holomorphic
(complex) submanifold and M⊥ is a totally real submanifold of a Kaehler manifold
M̄ . Then he introduced the notion of CR-warped products of Kaehler manifolds as
follows: A submanifold of a Kaehler manifold is called a CR-warped product if it
is the warped product MT ×f M⊥ of a holomorphic submanifold MT and a totally
real submanifold M⊥ . He also established a sharp relationship between the warping
function f of a warped product CR-submanifold MT ×f M⊥ of a Kaehler manifold
M̄ and the squared norm of the second fundamental form k h k2 . After Chen’s
papers, CR-warped product submanifolds have been studied in various manifolds
[3, 19] and [24].
In [23], we proved that there are no warped product semi-slant (in the sense of
Papaghiuc) submanifolds. In [25], we studied hemi-slant submanifolds and showed
that there exists a class of warped product hemi-slant submanifolds.
In this paper, we introduce and study warped product skew CR-submanifolds
of Kaehler manifolds. One can observe that such skew CR-submanifolds are generalization of CR-warped products and warped product hemi-slant submanifolds of
Kaehler manifolds. Our construction of skew CR-warped submanifolds can be considered as a special multiply warped product manifolds. We note that multiply
warped product manifolds were studied in [12] and they were applied to spacetime
geometry. For example, in [12] and [13], the authors showed a representation of
the interior Schwarzschild space-time as a multiply warped product space-time with
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Skew CR-warped products
certain warping functions. Also, Dobarro and Ünal obtained in [15] that ReissnerNordström space-time and Kasner spacetime can be expressed as a multiply warped
product space-time. In sections 3-4 of this paper, we give an example of warped
product skew CR-submanifolds, obtain a characterization and establish an inequality which is a generalization of a sharp inequality obtained by Chen in [9].
2. Preliminaries
Let (M̄ , g) be a Kaehler manifold . This means [26] that M̄ admits a tensor field J
of type (1,1) on M̄ such that, ∀X, Y ∈ Γ(T M̄ ), we have
J 2 = −I,
g(X, Y ) = g(JX, JY ),
¯ X J)Y = 0,
(∇
(2)
¯ is the Levi-Civita connection on M̄ . Let
where g is the Riemannian metric and ∇
M be a Riemannian manifold isometrically immersed in M̄ and denote by the same
symbol g for the induced Riemannian metric on M. Let Γ(T M ) be the Lie algebra
of vector fields in M and Γ(T M ⊥ ) the set of all vector fields normal to M. Denote
by ∇ the Levi-Civita connection of M. Then the Gauss and Weingarten formulas
are given by
¯ X Y = ∇X Y + h(X, Y )
∇
(3)
and
¯ X N = −AN X + ∇⊥
∇
XN
⊥
(4)
⊥
for any X, Y ∈ Γ(T M ) and any N ∈ Γ(T M ), where ∇ is the connection in
the normal bundle T M ⊥ , h is the second fundamental form of M and AN is the
Weingarten endomorphism associated with N. The second fundamental form h and
the shape operator A are related by
g(AN X, Y ) = g(h(X, Y ), N ).
(5)
For any X ∈ Γ(T M ) we write
JX = T X + F X,
(6)
where T X is the tangential component of JX and F X is the normal component of
JX. Similarly, for any vector field normal to M , we put
JN = BN + CN,
(7)
where BN and CN are the tangential and the normal components of JN , respectively.
Let M̄ be a Kaehler manifold with complex structure J and M is a Riemannian
manifold isometrically immersed in M̄ . Then M is called holomorphic (complex) if
J(Tp M ) ⊂ Tp M for every p ∈ M where Tp M denotes the tangent space to M at the
point p. M is called totally real if J(Tp M ) ⊂ Tp M ⊥ for every p ∈ M, where Tp M ⊥
denotes the normal space to M at the point p. Besides holomorphic and totally real
submanifolds, there are several other classes of submanifolds of a Kaehler manifold
determined by the behavior of the tangent bundle of the submanifold under the
action of the complex structure of the ambient manifold.
192
B. S.ahin
(1) The submanifold M is called a CR-submanifold [1] if there exists a differentiable distribution D : p → Dp ⊂ Tp M such that D is invariant with respect
to J and the complementary distribution D⊥ is anti-invariant with respect to
J.
(2) The submanifold M is called slant [8] if for all non-zero vector X tangent to
M the angle θ(X) between JX and Tp M is a constant, i.e. it does not depend
on the choice of p ∈ M and X ∈ Tp M.
(3) The submanifold M is called semi-slant [21] if it is endowed with two orthogonal
distributions DT and Dθ , where DT is invariant with respect to J and Dθ is
slant, i.e. θ(X) between JX and Dpθ is constant for X ∈ Dpθ .
(4) The submanifold M is called a hemi-slant submanifold ([6] and[25]) if it is
endowed with two orthogonal distributions Dθ and D⊥ , where Dθ is slant and
¯
D⊥ is anti-invariant with respect to J.
Finally, we recall the definition of skew CR-submanifolds from [22]. Let M
be a submanifold of a Kaehler manifold M̄ . For any Xp and Yp in Tp M we have
g(T Xp , Yp ) = −g(Xp , T Yp ). Hence, it follows that T 2 is a symmetric operator on the
tangent space Tp M, for all p. Therefore, its eigenvalues are real and diagonalizable.
Moreover, its eigenvalues are bounded by −1 and 0. For each p ∈ M, we may set
Dpλ = Ker{T 2 + λ2 (p)I}p ,
where I is the identity transformation and λ(p) belongs to closed real interval [0, 1]
such that −λ2 (p) is an eigenvalue of T 2 (p). Notice that Dp1 = KerF and Dp0 = KerT.
Dp1 is the maximal J− invariant subspace of Tp M and Dp0 is the maximal anti J− invariant subspace of Tp M. From now on, we denote the distributions D1 and D0 by DT
and D⊥ , respectively. Since Tp2 is symmetric and diagonalizable, if −λ21 (p), ..., −λ2k (p)
are the eigenvalues of T 2 at p ∈ M, then Tp M can be decomposed as a direct sum
of mutually orthogonal eigenspaces, i.e.
Tp M = Dpλ1 ⊕ ... ⊕ Dpλk .
Each Dpλi , 1 ≤ i ≤ k, is a T − invariant subspace of Tp M. Moreover, if λi 6= 0, then
Dpλi is even dimensional. Let M be a submanifold of a Kaehler manifold M̄ . M is
called a generic submanifold if there exists an integer k and functions λi , 1 ≤ i ≤ k
defined on M with values in (0, 1) such that
1. Each −λ2i (p), 1 <≤ i ≤ k is a distinct eigenvalue of T 2 with
Tp M = DpT ⊕ Dp⊥ ⊕ Dpλ1 ⊕ ... ⊕ Dpλk
for p ∈ M.
2. The dimensions of DpT , Dp⊥ and Dλi , 1 ≤ i ≤ k are independent of p ∈ M.
Skew CR-warped products
193
Moreover, if each λi is constant on M , then M is called a skew CR-submanifold.
Thus, we observe that CR-submanifolds are a particular class of skew CR-submanifolds with k = 0, DT 6= {0} and D⊥ 6= {0}. And slant submanifolds are also a
particular class of skew CR-submanifolds with k = 1, DT = {0}, D⊥ = {0} and
λ1 is constant. Moreover, if D⊥ = {0}, DT 6= {0} and k = 1, then M is a semislant submanifold. Furthermore, if DT = {0}, D⊥ 6= {0} and k = 1, then M is a
hemi-slant submanifold.
Definition 1. We say that a submanifold M is a skew CR-submanifold of order 1
if M is a skew CR-submanifold with k = 1 and λ1 is constant.
Thus a slant submanifold is a skew CR-submanifold of order 1 with DT = {0}
and D⊥ = {0}. Moreover, a semi-slant submanifold is a skew CR-submanifold of
order 1 with D⊥ = {0} and a hemi-slant submanifold is a skew CR-submanifold of
order 1 with DT = {0}. We say that a skew CR-submanifold of order 1 is proper if
DT 6= {0} and D⊥ 6= {0}.
Recall that slant submanifolds are characterized by
T 2 X = λX
(8)
such that λ ∈ [−1, 0], for details see [8]). If M is slant, then λ = −cos2 θ, where θ
is the slant angle of M. Using (8), we also have another characterization for slant
submanifolds. Namely, M is a slant submanifold if and only if there exists a constant
κ ∈ [−1, 0] such that
BF X = κX
(9)
for X ∈ Γ(T M ). If M is a slant submanifold, then BF X = −sin2 θ X [25]. In this
case, we also have
CF X = −F T X.
(10)
3. Warped product skew CR-submanifolds in Kaehler manifolds
Let M1 be a semi-slant submanifold in the sense of Papaghiuc and M⊥ a totally real
submanifold of a Kaehler manifold. In this section we consider a warped product
M = M1 ×f M⊥ in a Kaehler manifold M̄ . Then, it is obvious that M is a proper
skew CR-submanifold of order 1 of M̄ . So, we are entitled to call M a skew CRwarped product of M̄ . Then, from definition of a semi-slant submanifold and a skew
CR-submanifold, we have
T M = Dθ ⊕ DT ⊕ D⊥ ,
(11)
where Dpθ = Dpλ1 . Thus, if Dθ = {0}, then M is a CR-warped product [9]. If
DT = {0}, then M becomes a warped product hemi-slant submanifold [25].
Remark 1. From [9, Theorem 3.1], we know that there are no proper warped product CR-submanifolds in the form M⊥ ×f MT of a Kaehler manifold M̄ such that
M⊥ is totally real and MT is a holomorphic submanifold in M̄ . Also, from [25,
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B. S.ahin
Theorem 4.2], we know that there are no proper warped product submanifolds in the
form M⊥ ×f Mθ of a Kaehler manifold M̄ such that M⊥ is totally real and Mθ is
a proper slant submanifold in M̄ . Thus, it follows that there are no warped product
skew CR-submanifolds of order 1 in the form form M⊥ ×f M1 of a Kaehler manifold
M̄ such that M⊥ is totally real and M1 is a semi-slant submanifold (in the sense of
Papaghiuc) in M̄ .
We first give an example of a warped product skew CR-submanifold.
Example 1. Consider a submanifold in R12 given by the following equations:
x1 = u1 cos θ, x2 = u2 , x3 = u1 sin θ, x4 = 0,
x5 = u3 cos u5 , x6 = −u4 cos u5 , x7 = u3 sin u5 , x8 = −u4 sin u5
x9 = u1 sin u5 , x10 = 0 , x11 = u1 cos u5 , x12 = 0.
Then, the tangent bundle of M is spanned by
∂
∂
∂
∂
∂
+ sin θ
+ sin u5
+ cos u5
, Z2 =
∂ x1
∂ x3
∂ x9
∂ x11
∂ x2
∂
∂
∂
∂
Z3 = cos u5
+ sin u5
, Z4 = −cos u5
− sin u5
∂ x5
∂ x7
∂ x6
∂ x8
∂
∂
∂
∂
+ u4 sin u5
+ u3 cos u5
− u4 cos u5
Z5 = −u3 sin u5
∂ x5
∂ x6
∂ x7
∂ x8
∂
∂
+u1 cos u5
− u1 sin u5
.
∂ x9
∂ x11
Z1 = cos θ
Then it is easy to see that Dθ = span{Z1 , Z2 } is a slant distribution with slant an√ θ and D T = span{Z3 , Z4 } is a holomorphic distribution.
gle ϕ such that cos ϕ = cos
2
¯ 5 is orthogonal to M. Thus D⊥ = span{Z5 }
Moreover, it can be easily seen that JZ
is an anti-invariant distribution. Hence, we conclude that M is a proper skew CRsubmanifold of order 1 of R12 . Moreover, it is easy to see that DT ⊕ Dθ and D⊥ are
integrable. Denoting the integral manifolds of DT , Dθ and D⊥ by MT , Mθ and M⊥ ,
respectively, then the induced metric tensor of M is
ds2 = 2du21 + du22 + du23 + du24 + (u23 + u24 + u21 )du25
= gMθ + gMT + (u23 + u24 + u21 )gM⊥ .
12
Thus, it follows that
p M is a warped product skew CR-submanifold of R with warp2
2
2
ing function f = u3 + u4 + u1 .
In the rest of this section, we are going to give a characterization of a skew
CR-warped product of a Kaehler manifold M̄ .
Lemma 1. Let M be a proper skew CR-submanifold of order 1 of a Kaehler manifold
M̄ . Then, we have
g(∇X1 Y1 , Z) = g(AJZ X1 , JY1 )
g(∇X1 Y2 , Z) = sec2 θ{−g(AF T Y2 X1 , Z) + g(AJZ T Y2 , X1 )}
(12)
(13)
g(∇Y2 X1 , Z) = g(AJZ Y2 , JX1 )
(14)
for X1 , Y1 ∈ Γ(DT ), Y2 ∈ Γ(Dθ ) and Z ∈ Γ(D⊥ ).
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Skew CR-warped products
¯ X JY1 , JZ) for X1 , Y1 ∈
Proof. From (3) and (2), we obtain g(∇X1 Y1 , Z) = g(∇
1
T
⊥
Γ(D ) and Z ∈ Γ(D ). Then using again (3) and (5), we have (12). With regards
¯ X JY2 , JZ). Using
to statement (13), from (3) and (2), we get g(∇X1 Y2 , Z) = g(∇
1
(6) we derive
¯ X T Y2 , JZ) + g(∇
¯ X F Y2 , JZ).
g(∇X1 Y2 , Z) = g(∇
1
1
Thus, from (3) and (2) we have
¯ X Z).
g(∇X1 Y2 , Z) = g(h(X1 , T Y2 ), JZ) + g(JF Y2 , ∇
1
Hence, using (7), we get
g(∇X1 Y2 , Z) = g(h(X1 , T Y2 ), JZ) + g(BF Y2 , ∇X1 Z) + g(CF Y2 , h(X1 , Z)).
Then, from (9) and (10), we arrive at
g(∇X1 Y2 , Z) = g(h(X1 , T Y2 ), JZ) − sin2 θ g(Y2 , ∇X1 Z) − g(F T Y2 , h(X1 , Z)).
Hence, we have
g(∇X1 Y2 , Z) = g(h(X1 , T Y2 ), JZ) + sin2 θ g(∇X1 Y2 , Z) − g(F T Y2 , h(X1 , Z)).
Thus, using (5) we obtain (13). In a similar way, one can obtain (14).
Lemma 2. Let M be a proper skew CR-submanifold of order 1 of a Kaehler manifold
M̄ . Then, we have
g(∇X2 Y2 , Z) = g(AJZ T Y2 − AF T Y2 Z, X2 )sec2 θ
g(∇Z V, X2 ) = −sec2 θ {g(AJV Z, T X2 ) − g(AF T X2 Z, V )}
(15)
(16)
g(∇Z V, X1 ) = −g(AJV Z, JX1 )
(17)
and
T
θ
⊥
for any X1 ∈ Γ(D ), X2 , Y2 ∈ Γ(D ) and Z, V ∈ Γ(D ).
Proof. The proof of (17) is exactly the same as in (12). For the proof of (15),
¯ X T Y2 , JZ) + (∇
¯ X F Y2 , JZ) for
from (2), (3) and (6), we have g(∇X2 Y2 , Z) = g(∇
2
2
θ
⊥
X2 , Y2 ∈ Γ(D ) and Z, V ∈ Γ(D ). Then using (4) and (2) we get
¯ X JF Y2 , Z).
g(∇X2 Y2 , Z) = (h(X2 , T Y2 ), JZ) − g(∇
2
Thus, using (7) we derive
¯ X BF Y2 , Z) − g(∇
¯ X CF Y2 , Z).
g(∇X2 Y2 , Z) = (h(X2 , T Y2 ), JZ) − g(∇
2
2
Hence, we obtain
¯ X F T Y2 , Z).
g(∇X2 Y2 , Z) = (h(X2 , T Y2 ), JZ) + sin2 θ g(∇X2 Y2 , Z) + g(∇
2
Then, from (4) we have
g(∇X2 Y2 , Z) = (h(X2 , T Y2 ), JZ) + sin2 θ g(∇X2 Y2 , Z) − g(AF T Y2 X2 , Z).
Thus, using again (5) we obtain (15). In a similar way, one can obtain (16).
196
B. S.ahin
Lemma 3. Let M be a proper skew CR-submanifold of order 1 of a Kaehler manifold
M̄ . Then we have
sin2 θg(∇Z X1 , Y2 ) = −g(AF T Y2 Z, X1 ) + g(AF Y2 Z, JX1 )
for X1 ∈ Γ(DT ), Y2 ∈ Γ(Dθ ) and Z ∈ Γ(D⊥ ).
Proof. From (3), (2) and (6) we obtain
¯ Z JX1 , T Y2 ) + g(∇
¯ Z JX1 , F Y2 )
g(∇Z X1 , Y2 ) = g(∇
for X1 ∈ Γ(DT ), Y2 ∈ Γ(Dθ ) and Z ∈ Γ(D⊥ ). Then, using (2), (6) and (3) we get
¯ Z X1 , T 2 Y2 ) − g(∇
¯ Z X1 , F T Y2 ) + g(h(Z, JX1 ), F Y2 ).
g(∇Z X1 , Y2 ) = −g(∇
Thus, from (8) we have
g(∇Z X1 , Y2 ) = cos2 θ g(∇Z X1 , Y2 ) − g(h(Z, X1 ), F T Y2 ) + g(h(Z, JX1 ), F Y2 ).
Hence, using (5) we obtain the assertion of lemma.
We now recall the following definition from [22]. For the distributions Dθ and
D on M, we say that M is Dθ − D⊥ mixed totally geodesic if for all X ∈ Γ(Dθ )
and Y ∈ Γ(D⊥ ), we have h(X, Y ) = 0.
We also recall the result of S. Hiepko [18], (cf. [14], Remark 2.1): Let D1 be a
vector subbundle in the tangent bundle of a Riemannian manifold M and let D2 be
its normal bundle. Suppose that the two distributions are involutive. We denote
the integral manifolds of D1 and D2 by M1 and M2 , respectively. Then M is locally
isometric to warped product M1 ×f M2 if the integral manifold M1 is totally geodesic
and the integral manifold M2 is an extrinsic sphere, i.e. M2 is a totally umbilical
submanifold with a parallel mean curvature vector.
Now, we are ready to state and prove a characterization for a skew CR-warped
product in a Kaehler manifold.
⊥
Theorem 1. Let M be a Dθ −D⊥ mixed totally geodesic proper skew CR-submanifold
of order 1 of a Kaehler manifold M̄ . Then M is a locally skew CR-warped product
if and only if the following holds:
(a) AJZ T Y2 has no components in Dθ for Z ∈ Γ(D⊥ ) and Y2 ∈ Γ(Dθ ).
(b) For X1 ∈ Γ(DT ), Z ∈ Γ(D⊥ ), X2 ∈ Γ(Dθ ) and a function µ defined on M, we
have
AJZ JX1 = X1 (µ)Z
AF T X2 Z = −X2 (µ)cos2 θZ
such that V (µ) = 0 for every V ∈ Γ(D⊥ ).
(18)
(19)
Skew CR-warped products
197
Proof. Let M be a Dθ − D⊥ mixed totally geodesic skew CR warped product of a
Kaehler manifold M̄ . Then M1 is totally geodesic in M. Thus, ∇X Y ∈ Γ(T M1 ) for
X, Y ∈ Γ(T M1 ). On the other hand, Dθ − D⊥ mixed totally geodesic M and (15)
imply that
g(∇X2 Y2 , Z) = sec2 θ g(AJZ T Y2 , X2 )
for X2 , Y2 ∈ Γ(Dθ ) and Z ∈ Γ(D⊥ ). Then, since ∇X2 Y2 ∈ Γ(T M1 ), we have
g(AJZ T Y2 , X2 ) = 0
which implies (a). Moreover, from (12) and (14) we obtain
g(AJZ X1 , JY1 ) = 0 and
g(AJZ Y2 , JX1 ) = 0
for X1 , Y1 ∈ Γ(DT ), Y2 ∈ Γ(Dθ ) and Z ∈ Γ(D⊥ ). Since A is self-adjoint, we conclude
that AJZ X1 has no components in T M1 . Then, AJZ JX1 ∈ Γ(D⊥ ). Taking into
account that M is Dθ − D⊥ mixed totally geodesic, from (1) and (17) we obtain
X1 (lnf )g(Z, V ) = g(AJV Z, JX1 )
for V ∈ Γ(D⊥ ). Hence, we obtain (18). Since M is Dθ − D⊥ mixed totally geodesic,
we have g(AF T Y2 Z, Y2 ) = 0. Hence, AF T Y2 Z has no components in Dθ . On the other
hand, (18) implies that g(AJV X1 , Y2 ) = −JX1 (µ)g(V, Y2 ) = 0. Then, using (1) and
(13), g(AF T Y2 Z, X1 ) = 0 which implies that AF T Y2 Z has also no components in DT .
Then, from (11), we conclude that AF T Y2 Z ∈ Γ(D⊥ ). Thus, mixed totally geodesic
Dθ − D⊥ and (16) imply that
g(∇Z V, Y2 ) = g(AF T Y2 Z, V ) sec2 θ.
Then, using (1) we get
g(AF T Y2 Z, V ) = −cos2 θY2 (lnf )g(Z, V )
which is (19). Moreover, since Z(lnf ) = 0 for a skew CR-product, we obtain µ = lnf.
Let us to prove converse. Assume that M is Dθ − D⊥ mixed totally geodesic
proper skew CR-submanifold of order 1 of a Kaehler manifold M̄ such that (a) and
(b) hold. Then, from (a), (b) and (12)-(15), it is easy to see that M1 is totally
geodesic in M. On the other hand, from [7], D⊥ is always integrable. We denote
the integral manifold of D⊥ by M⊥ . Also let h⊥ be the second fundamental form of
M⊥ in M . From (3) we have g(h⊥ (Z, V ), X1 ) = g(∇Z V, X1 ) for X1 , ∈ Γ(DT ) and
Z, V ∈ Γ(D⊥ ). Then, (17) implies that g(h⊥ (Z, V ), X1 ) = −g(AJV Z, JX1 ). Thus,
from (18) we obtain
g(h⊥ (Z, V ), X1 ) = −X1 (µ)g(V, Z).
(20)
In a similar way, from (3) we get g(h⊥ (Z, V ), X2 ) = g(∇Z V, X2 ) for X2 ∈ Γ(Dθ ).
Then,since M is Dθ −D⊥ mixed totally geodesic, from (16) we obtain g(h⊥ (Z, V ), X2 )
= g(AF T X2 Z, V )sec2 θ. Using (19) we have
g(h⊥ (Z, V ), X2 ) = −X2 (µ)g(Z, V ).
(21)
198
B. S.ahin
Thus, for X = X1 + X2 ∈ Γ(M1 ), from (20) and (21) we obtain
g(h⊥ (Z, V ), X) = g(h⊥ (Z, V ), X1 ) + g(h⊥ (Z, V ), X2 )
= −[X1 (µ) + X2 (µ)]g(Z, V )
(22)
which implies that M⊥ is totally umbilical in M. Denoting the gradient of µ on DT
and Dθ by gradT µ and gradθ µ, respectively, from (22) we can write
h⊥ (Z, V ) = −[gradT µ + gradθ µ]g(Z, V ).
(23)
Thus, for X = X1 + X2 ∈ Γ(T M1 ), we have
g(∇Z gradT µ + gradθ µ, X) = g(∇Z gradT µ, X) + g(∇Z gradθ µ, X)
= [Zg(gradT µ, X2 ) − g(gradT µ, ∇Z X)]
+Zg(gradθ µ, X1 ) − g(gradθ µ, ∇Z X)
= [Z(X2 (µ)) − g(gradT µ, ∇Z X)]
+Z(X1 (µ)) − g(gradθ µ, ∇Z X).
Then, using (19) in Lemma 3.3, we get g(∇Z X1 , X2 ) = −g(∇Z X2 , X1 ) = 0. Hence,
we arrive at
g(∇Z gradT µ + gradθ µ, X) = [Z(X2 (µ)) − g(gradT µ, ∇Z X2 )]
+Z(X1 (µ)) − g(gradθ µ, ∇Z X1 ).
Hence, by direct calculations, we get
g(∇Z gradT µ + gradθ µ, X) = [Z(X2 (µ)) − [Z, X2 ](µ)
+g(gradT µ, ∇X2 Z)] + Z(X1 (µ))
−[Z, X1 ](µ) + g(gradθ µ, ∇X1 Z).
= [X2 (Z(µ)) + g(gradT µ, ∇X2 Z)]
−X1 (Z(µ)) + g(gradθ µ, ∇X1 Z).
(24)
Since Z(µ) = 0, we derive
g(∇Z gradT µ + gradθ µ, X) = −g(∇X2 gradT µ, Z) − g(∇X1 gradθ µ, Z).
On the other hand, since M1 is totally geodesic, we have ∇X1 gradθ µ, ∇X2 gradT µ ∈
Γ(T M1 ). Hence, we obtain
g(∇Z gradT µ + gradθ µ, X) = 0.
Thus, we conclude that gradT µ+gradθ µ is parallel in M. This result and (23) imply
that M⊥ is an extrinsic sphere. Thus, the proof is complete.
4. An inequality for skew CR-warped products
In this section, we obtain an inequality for the squared norm of the second fundamental form in terms of the warping function for skew CR-warped products in
Kaehler manifolds. To do this, we need the following lemmas.
Skew CR-warped products
199
Lemma 4. Let M be a skew CR-warped product of a Kaehler manifold M̄ . Then,
we have
g(h(X1 , Y2 ), JZ) = 0
(25)
and
g(h(X1 , Y1 ), JZ) = 0
T
θ
(26)
⊥
for X1 , Y1 ∈ Γ(D ), Y2 ∈ Γ(D ) and Z ∈ Γ(D ).
Proof. From (2) and (3) we get
¯ Y X1 , JZ)
g(∇Y2 JX1 , Z) = −g(∇
2
for X1 ∈ Γ(DT ), Y2 ∈ Γ(Dθ ) and Z ∈ Γ( D⊥ ). Hence, we have g(JX1 , ∇Y2 Z)
¯ Y X1 , JZ). Then, using (1) and (3) we obtain
= g(∇
2
Y2 (lnf )g(JX1 , Z) = g(h(X1 , Y2 ), JZ).
Hence, g(h(X1 , Y2 ), JZ) = 0. In a similar way, from (3) and (2) we get g(∇X1 Z, Y1 )
¯ X JZ, JY1 ) for X1 , Y1 ∈ Γ(DT ) and Z ∈ Γ(D⊥ ). Then, using (1) and (4) we
= g(∇
1
obtain 0 = g(AJZ X1 , JY1 ). Thus (3) implies (26).
Lemma 5. Let M be a skew CR-warped product of a Kaehler manifold M̄ . Then,
we have
X1 (lnf )g(Z, V ) = g(h(Z, JX1 ), JV )
(27)
and
g(h(Z, X2 ), F Y2 ) = g(h(X2 , Y2 ), JZ)
T
θ
(28)
⊥
for X1 ∈ Γ(D ), X2 , Y2 ∈ Γ(D ) and Z, V ∈ Γ(D ).
Proof. Equation (27) was obtained in [9], Lemma 4.1(3). For (28), from (3), we
¯ X Z, F T Y2 ) for X2 , Y2 ∈ Γ(Dθ ) and Z, V ∈ Γ(D⊥ ).
have g(h(X2 , Z), F T Y2 ) = g(∇
2
¯ X Z, JT Y2 − T 2 Y2 ). Here, (8) and
Then, using (6) we get g(h(X2 , Z), F T Y2 ) = g(∇
2
(2) imply that
¯ X JZ, T Y2 ).
g(h(X2 , Z), F T Y2 ) = cos2 θ g(∇X2 Z, Y2 ) − g(∇
2
Thus, from (1) and (4), we have
g(h(X2 , Z), F T Y2 ) = g(AJZ X2 , T Y2 ).
Then, using (5) we get
g(h(X2 , Z), F T Y2 ) = g(h(X2 , T Y2 ), JZ).
Thus, substituting T Y2 by Y2 and using (8) we obtain (28).
Lemma 6. Let M be a skew CR-warped product of a Kaehler manifold M̄ . Then,
we have
T X2 (lnf )g(Z, V ) = −g(h(Z, X2 ), JV ) + g(h(Z, V ), F X2 )
(29)
for X2 ∈ Γ(Dθ ), and Z, V ∈ Γ(D⊥ ).
200
B. S.ahin
Proof. From (2), (3) and (6) we get
¯ Z T X2 , JV ) + g(∇
¯ Z F X2 , JV )
g(∇X2 Z, V ) = g(∇
for X2 ∈ Γ(Dθ ), and Z, V ∈ Γ(D⊥ ). Hence, using (1) and (3), we obtain
¯ Z JV ).
X2 (lnf )g(Z, V ) = g(h(Z, T X2 ), JV ) − g(F X2 , ∇
Using again (2) we arrive at
¯ Z V ).
X2 (lnf )g(Z, V ) = g(h(Z, T X2 ), JV ) + g(JF X2 , ∇
Thus, from (7), we derive
¯ Z V ).
X2 (lnf )g(Z, V ) = g(h(Z, T X2 ), JV ) + g(BF X2 + CF X2 , ∇
Then, (3), (4), (9) and (10) imply that
X2 (lnf )g(Z, V ) = g(h(Z, T X2 ), JV ) − sin2 θg(X2 , ∇Z V ) − g(F T X2 , h(Z, V )).
Hence, we have
X2 (lnf )g(Z, V ) = g(h(Z, T X2 ), JV ) + sin2 θg(∇Z X2 , V ) − g(F T X2 , h(Z, V )).
Thus, using(1) we obtain
X2 (lnf )g(Z, V ) = g(h(Z, T X2 ), JV ) + sin2 θX2 (lnf )g(Z, V ) − g(F T X2 , h(Z, V )).
Thus, we get
cos2 θ X2 (lnf )g(Z, V ) = g(h(Z, T X2 ), JV ) − g(F T X2 , h(Z, V )).
Now, substituting X2 by T X2 and using (8), we obtain(29).
Let M be an (m1 + m2 + m3 )− dimensional proper skew CR-warped product
of a Kaehler manifold M̄ m2 +m3 . Then, we choose a canonical orthonormal frame
{e1 , ..., em1 , ē1 , ..., ēm2 , e˜1 , ..., ẽm3 , J e˜1 , ..., J ẽm3 , e∗1 , ..., e∗m2 } such that {e1 , ..., em1 }
is an orthonormal basis of DT , {ē1 , ..., ēm2 } is an orthonormal basis of Dθ and
{e˜1 , ..., ẽm3 } is an orthonormal basis of D⊥ . It is known that the ranks of DT and
Dθ are even. Hence, m1 = 2n1 and m2 = 2n2 . Then, we can choose orthonormal
frames {ē1 , ..., ēm2 } and{e∗1 , ..., e∗m2 } such that
e¯2 = sec θ T ē1 , . . .ē2n2 = sec θ T ē2n2 −1 ,
e∗1 = csc θ F ē1 , . . .e∗2n2 = csc θ F ē2n2 ,
where θ is the slant angle of Dθ . Notice that it is called such an orthonormal frame
an adapted frame [8].
We are now ready to state and prove the main theorem of this section.
Theorem 2. Let M be an (m1 + m2 + m3 )− dimensional proper Dθ − D⊥ mixed
totally geodesic skew CR-warped product of a Kaehler manifold M̄ m2 +m3 . Then we
have
201
Skew CR-warped products
(1) The squared norm of the second fundamental form of M satisfies
k h k2 ≥ m3 [2 k ∇T (lnf ) k2 +cot2 θ k ∇θ (lnf ) k2 ],
(30)
where m3 = dim(M⊥ ), ∇T (lnf ) and ∇θ (lnf ) are gradients of lnf on DT and
Dθ , respectively.
(2) If the equality sign of (30) holds identically, then M1 is a totally geodesic
submanifold and M⊥ is a totally umbilical submanifold of M̄ .
Proof. Since
k h k2 = k h(DT , DT ) k2 + k h(Dθ , Dθ ) k2 + k h(D⊥ , D⊥ ) k2 +2 k h(DT , D⊥ ) k2
+2 k h(DT , Dθ ) k2 +2 k h(D⊥ , Dθ ) k2
and M is Dθ − D⊥ mixed totally geodesic, we get
k h k2 = k h(DT , DT ) k2 + k h(Dθ , Dθ ) k2 + k h(D⊥ , D⊥ ) k2 +2 k h(DT , D⊥ ) k2
+2 k h(DT , Dθ ) k2 .
Hence, we have
m1 X
m3
X
2
khk =
g(h(ei , ej ), J ẽa ) +
i,j=1 a=1
m3
X
+
+
2
a,b,c=1
m2
X
g(h(ẽa , ẽb ), J ẽc )2 +
g(h(ēr , ēs ), e∗p )2 +
p,q,r=1
m1 X
m3 X
m2
X
+2
+2
m1 X
m2
X
g(h(ei , ej ), e∗r )2
i,j=1 r=1
m3 X
m2
X
g(h(ẽa , ẽb ), e∗r )2
a,b=1 r=1
m
m2
3 X
X
g(h(ēr , ēs ), J ẽa )2
a=1 r,s=1
2
g(h(ei , ēr ), J ẽa ) + 2
m1 X
m2
X
i=1 a=1 r=1
m1 X
m3
X
i=1 r,s=1
m
m3 X
m2
1
XX
i=1 a,b=1
i=1 a=1 r=1
g(h(ei , ẽa ), J ẽb )2 + 2
g(h(ei , ēr ), e∗s )2
g(h(ei , ẽa ), e∗r )2 .
Then, using (25), (26), (28), (27) and considering the adapted frame, we obtain
k h k2 =
m1 X
m2
X
g(h(ei , ej ), F ēr )2 csc2 θ +
i,j=1 r=1
+
m3 X
m2
X
m3
X
a,b,c=1
m2
X
g(h(ẽa , ẽb ), F ēr )2 csc2 θ +
a,b=1 r=1
m1 X
m2
X
+2
g(h(ēr , ēs ), F ēp )2 csc2 θ
p,q,r=1
g(h(ei , ēr ), F ēs )2 csc2 θ + 2
i=1 r,s=1
+2
g(h(ẽa , ẽb ), J ẽc )2
m1 X
m3 X
m2
X
i=1 a=1 r=1
m1 X
m3
X
i=1 a,b=1
g(h(ei , ẽa ), F ēr )2 csc2 θ.
[Jei (lnf )g(ẽa , ẽb )]2
202
B. S.ahin
Thus, Dθ − D⊥ mixed totally geodesic M and (29) imply that
k h k2 =
m1 X
m2
X
i,j=1 r=1
m2
X
+
m3
X
g(h(ei , ej ), F ēr )2 csc2 θ +
a,b,c=1
m3 X
m2
X
g(h(ēr , ēs ), F ēp )2 csc2 θ +
p,q,r=1
+2
m1 X
m2
X
[T ēr (lnf )g(ẽa , ẽb )]2 csc2 θ
a,b=1 r=1
m1 X
m3
X
g(h(ei , ēr ), F ēs )2 csc2 θ + 2
i=1 r,s=1
+2
g(h(ẽa , ẽb ), J ẽc )2
[Jei (lnf )g(ẽa , ẽb )]2
i=1 a,b=1
m1 X
m3 X
m2
X
g(h(ei , ẽa ), F ēr )2 csc2 θ.
(31)
i=1 a=1 r=1
On the other hand, by direct computations, we we have
m2
X
[T ēr (lnf )]2 csc2 θ = [T ē1 (lnf )]2 csc2 θ + [sec θT 2 ē1 (lnf )]2 csc2 θ
p=1
+[T ē2 (lnf )]2 csc2 θ + [sec θT 2 ē2 (lnf )]2 csc2 θ + . . .
+[T ē2n2 −1 (lnf )]2 csc2 θ + [sec θT 2 ē2n2 −1 (lnf )]2 csc2 θ
Then, using (8) we get
m2
X
[T ēr (lnf )]2 csc2 θ = [T ē1 (lnf )sec θ]2 cos2 θcsc2 θ + [−cos θē1 (lnf )]2 csc2 θ
p=1
+[T ē2 (lnf )sec θ]2 cos2 θcsc2 θ + [−cos θē2 (lnf )]2 csc2 θ
+ · · · + [T ē2n2 sec θ(lnf )]2 cos2 θcsc2 θ
+[−cos θē2n2 −1 (lnf )]2 csc2 θ.
Hence, we arrive at
m2
X
[T ēr (lnf )]2 csc2 θ = cot2 θ(∇θ lnf )2 .
(32)
p=1
Then, using (32) in (31) we obtain (30). If the equality sign of (30) holds, from (31)
we have
h(DT , DT ) = 0,
⊥
⊥
h(Dθ , DT ) = 0,
θ
h(D , D ) ⊂ F (D ),
θ
θ
h(DT , D⊥ ) ⊂ J(D⊥ ),
h(D , D ) ⊂ J(D⊥ ).
(33)
(34)
Since M is Dθ − D⊥ mixed totally geodesic, from (28) we have g(h(X2 , Y2 ), JZ) = 0
for X2 , Y2 ∈ Γ(Dθ ) and Z ∈ Γ(D⊥ ). Thus, using (34) we obtain h(Dθ , Dθ ) = 0. Then
(33) implies that M1 being a totally geodesic in M̄ due to M1 is totally geodesic
Skew CR-warped products
203
submanifold of M. Furthermore, the first equation of (34), (29) and Dθ − D⊥ mixed
totally geodesic M imply that
g(h(Z, V ), F X2 ) = T X2 (lnf )g(Z, V )
for X2 ∈ Γ(Dθ ) and Z, V ∈ Γ(D⊥ ). Hence, since M⊥ is totally umbilical in M,
it follows that M⊥ is a totally umbilical submanifold of M̄ . Thus, the proof is
complete.
Remark 2. In case Dθ = {0}, Theorem 2 coincides with Theorem 5.1 of [9]. Thus,
Theorem 2 is a generalization of Theorem 5.1 of [9].
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